Generalized Ziggurat Algorithm for Unimodal and Unbounded Probability Density Functions with Zest

TL;DR

This paper introduces a modified Ziggurat algorithm capable of efficiently generating random numbers from all unimodal, unbounded probability density functions, with a C++ library implementation called Zest that outperforms standard libraries.

Contribution

The paper presents a novel generalized Ziggurat algorithm for unbounded PDFs, along with an efficient C++ library, Zest, that supports custom distributions and outperforms existing random number generators.

Findings

Zest is faster than STL and Boost in most cases.

Supports a wide range of distributions including normal, exponential, and t-distributions.

Provides an efficient uniform floating-point RNG for unbounded tails.

Abstract

We present a modified Ziggurat algorithm that could generate a random number from all unimodal and unbounded PDFs. For PDFs that have unbounded density and/or unbounded support we use a combination of nonlinear mapping function and rejection sampling to generate a random number from the peak and/or the tail distribution. A family of mapping functions and their corresponding acceptance probability functions are presented (along with the criteria for their use and their efficiency) that could be used to generate random numbers from infinite tails and unbounded densities. The Zest library which is a C++ implementation of this algorithm is also presented. Zest can efficiently generate normal, exponential, cauchy, gamma, Weibull, log-normal, chi-squared, student's t and Fisher's f variates. The user can also define their custom PDF as a class and supply it as a template argument to our library's class without modifying any part of the library. Performance of Zest is compared against performance of random modules of (GCC's implementation of) Standard Template Library (STL) and Boost. The presented results show that Zest is faster than both in most cases, sometimes by a factor of more than 10. We also present a C++ implementation of a uniform floating-point random number generator (RNG) which is capable of producing all representable floating-point numbers in $[0,1)$ which will be used in the Ziggurat algorithm near unbounded peaks and tails. The common method of dividing a random integer by the range of the RNG can not produce random floating-point numbers with fully random fraction bits and very small random numbers. The presented uniform floating-point RNG is very efficient and in the case of producing double precision floating-point numbers it's even faster than simply multiplying a 64-bit integer by $2^{-64}$.